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A COMPLETE LOCAL DOMAIN WITH NO SMALL COHEN-MACAULAY MODULE
plain text edition, version 2
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Everything is over the complex numbers C. The background needed is:
rings, ideals, modules, localization, and the first definitions about
projective varieties. Everything else (depth, Cohen-Macaulay,
intersection numbers, Chern classes, Harder-Narasimhan filtrations) is
defined where it is first used, and the references at the end are
pointers, never a dependency of the proof.
CONTENTS
1. The question and the answer
2. The cone of a polarized surface
3. Balancing the slopes
4. The inequality forced by an ample divisor
5. The numerical obstruction
6. The surface
7. The counterexample
8. References
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1. THE QUESTION AND THE ANSWER
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1.1 Depth, dimension, Cohen-Macaulay
A ring is local if it has exactly one maximal ideal; write (R,m) and
k = R/m for the residue field. It is Noetherian if every ideal is
finitely generated, and complete if the natural map from R to the
inverse limit of R/m^n is an isomorphism.
Let M be a nonzero finitely generated module over a Noetherian local
ring (R,m). An element x in m is M-regular if "multiply by x" is an
injective map M -> M. A list x_1,...,x_t is an M-regular sequence if
each x_i is regular on M/(x_1,...,x_{i-1})M. Then
depth M := the maximal length of an M-regular sequence,
independent of all choices, and
dim M := dim(R/Ann(M)),
the Krull dimension of the support of M. Always depth M <= dim M.
M is Cohen-Macaulay (CM) if depth M = dim M. A small CM module over R
is a nonzero finitely generated module M with depth M = dim R. (The
word "small" contrasts with "big" CM modules, which may be infinitely
generated.)
Examples. A regular local ring is CM over itself; if R is CM and x in
m is a nonzerodivisor then R/(x) is CM. Hence C[[x,y,z]]/(f), for any
nonzero f, is a 2-dimensional CM ring. Over a 3-dimensional ring,
"small CM module" means "nonzero finite module of depth 3".
1.2 The question
Hochster asked in the early 1970s: does every complete Noetherian local
domain have a small CM module? Some answers are positive. In
dimension at most 2 the normalization of a complete local domain is a
finite CM module. For finitely generated graded domains of positive
characteristic there are graded small CM modules. In general the
answer is no.
Theorem A. There is a 3-dimensional complete Noetherian normal local
domain R containing C, with residue field C, such that every nonzero
finitely generated R-module M has depth M <= 2. In particular R has
no small CM module.
Theorem B. Let X be a smooth integral projective surface over C and
let H be an ample, globally generated divisor on X. Write K_X for the
canonical divisor of X, c_2(X) for the second Chern number of X, and
R(X,H) for the completed cone of the pair (X,H) (Section 2). If
R(X,H) has a small CM module then
K_X^2 - 2 c_2(X) <= 6 H^2. (1.1)
Theorem A is Theorem B applied to one surface for which (1.1) fails.
1.3 The idea in four sentences
A small CM module over a completed cone is a free module over the
power series ring, hence a vector bundle on the punctured cone. Push
that bundle onto the exceptional surface X: it becomes a torsion-free
sheaf P on X. Two inequalities then fight over one number attached
to P. The triviality of the bundle at infinity forces that number to
be >= 0 (Section 4), while semistability of the pieces of P forces it
to be <= 1/2 - (K_X^2 - 2c_2(X))/(12H^2) (Section 5); comparing the
two gives (1.1).
1.4 The same idea with numbers
For a coherent sheaf F of positive rank N on the plane P^2 put
v(F) = 2 ch_2(F)/N - (c_1(F)/N)^2, (1.2)
where c_1(F) is the integer coefficient of the line class and ch_2(F)
is the degree of the second Chern character. If F is a direct sum of
N line bundles O(a_1),...,O(a_N), then c_1(F)/N is the average of the
numbers a_i and v(F) is exactly the variance of the numbers a_i.
Variance is never negative, and it is at most 1/4 when all the a_i lie
in an interval of length 1. That is the whole mechanism: the
variance is nonnegative because it comes from a divisor (Section 4),
and it is small because semistability bounds it (Section 5).
1.5 What is new here
Earlier obstructions concern restricted classes of witnesses. If R
contains Q and is normal, trace shows that a CM finite extension
algebra would force R itself to be CM, so an extension algebra is a
special kind of witness. Ulrich sheaves and graded modules are also
special. The argument below uses only that some finite module of
depth 3 exists: no grading, no algebra structure, no semistability and
no Ulrich condition on the module. What replaces those hypotheses is
a comparison of coherent extensions across the exceptional divisor of
the cone (Lemma 4.1).
1.6 Plan of the paper
Section 2 builds the completed cone R of (X,H), a regular model W of
Spec R, and turns a small CM module into a torsion-free sheaf E on W.
Section 3 modifies E along the exceptional surface so that the
Harder-Narasimhan slopes of its restriction P lie in an interval of
length at most 1. Section 4 proves that triviality of the pushforward
of E away from the exceptional divisor forces v(f_*P) >= 0. Section
5 combines that with Bogomolov's inequality for the semistable factors
and deduces (1.1). Section 6 constructs a surface with
K_X^2 - 2c_2(X) > 6H^2. Section 7 assembles Theorems A and B and the
uncompleted form.
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2. THE CONE OF A POLARIZED SURFACE
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Fix for the rest of the paper:
X : a smooth integral projective surface over C,
H : an ample divisor on X that is globally generated,
h : H^2, a positive integer,
D : the plane P^2.
2.1 A finite map to the plane
Lemma 2.1. There are sections s_0,s_1,s_2 of O_X(H) with no common
zero. They define a morphism f: X -> D with f^*O_D(1) = O_X(H); this
morphism is finite and surjective, and deg f = h.
Proof. Choose s_0 != 0. Its zero divisor has finitely many irreducible
components; choose s_1 not vanishing identically on any of them. Then
the zero divisors of s_0 and s_1 share no component, so their
intersection Z is finite. Choose s_2 not vanishing at any point of Z.
Now s_0,s_1,s_2 have no common zero, and the ratio
p |-> [s_0(p):s_1(p):s_2(p)] defines f.
X is projective, so f is proper. A fiber of positive dimension would
contain a curve contracted to a point, and then
H·C = (f^*O_D(1))·C = 0, contradicting ampleness of H. So f is finite.
For a finite map of degree e, the pullback formula for intersection
numbers gives
h = H^2 = (f^*O_D(1))^2 = e (O_D(1))^2 = e.
QED
2.2 The ring and the model
Let S = C[x_0,x_1,x_2], where x_i acts on sections as s_i, and let
T = T(X,H) = direct sum over j >= 0 of H^0(X, O_X(jH)),
a graded ring containing S. Let A = C[[x_0,x_1,x_2]] and let
n = T_{>0} be the irrelevant ideal. The completed cone of (X,H) is
R = R(X,H) = T (tensor over S) A,
which is also the completion of T at n; that identification is step (c)
of Proposition 2.3.
To resolve Spec R we use the following two schemes. Let V_0 be the
total space of O_D(-1), which is also the blowup of A^3 = Spec S at the
origin, with exceptional divisor D. Let W_0 = X x_D V_0, the total
space of O_X(-H). Base change along Spec S -> Spec A:
V = V_0 x_S Spec A, W = W_0 x_S Spec A.
Lemma 2.2. With this notation:
(i) W_0 -> Spec T is an isomorphism over the complement of the vertex,
and W_0 -> X is a line bundle;
(ii) g: W -> V is finite, p: W -> Spec R is proper, and p is an
isomorphism over the complement of the closed point of Spec R;
(iii) X in W and D in V are effective Cartier divisors with ideal
sheaves J and I satisfying
J = g^*I, I|_D = O_D(1), J|_X = O_X(H);
(iv) V and W are regular and integral;
(v) the ring of global functions on W is R.
Proof. (i) On X_i = {s_i != 0} the line bundle O_X(H) is trivialized by
s_i, and clearing poles along div(s_i) identifies
T[1/s_i] = O(X_i)[s_i, s_i^{-1}].
Both the punctured line bundle over X_i and the corresponding open part
of Spec T have this ring of functions, so W_0 -> Spec T is an
isomorphism over the complement of the vertex. Also the global
functions on the total space of a line bundle L are the sum of the
H^0 of all powers of L; for L = O_X(-H) this sum is T.
(ii) Spec T -> Spec S is finite, so W_0 -> V_0 is finite by base
change, and hence g: W -> V is finite. V -> Spec A is proper, being a
blowup, so W -> Spec A is proper; Spec R -> Spec A is separated, hence
p: W -> Spec R is proper. Over the punctured base, p and V -> Spec A
are the open immersions described in (i), by flat base change.
(iii) X in W_0 is the zero section of a line bundle, hence an effective
Cartier divisor, and base change preserves that. The ideal I of D in V
satisfies I|_D = O_D(1), because the normal bundle of the exceptional
divisor of a blowup is O_D(-1). Base change gives J = g^*I, and the
normal bundle of the zero section of the total space of O_X(-H) is
O_X(-H), so J|_X = O_X(H).
(iv) V is the blowup of A^3 at the origin, base changed to the
completion, hence regular. Let w be a closed point of W and let t be a
local equation of X at w. Then t is a nonzerodivisor on O_{W,w} and
O_{W,w}/(t) is the local ring of X at a closed point, which is regular.
Lifting generators of the quotient maximal ideal to O_{W,w} shows that
the embedding dimension of O_{W,w} is at most its dimension: there is
one more generator and, t being a nonzerodivisor, one more dimension.
Hence O_{W,w} is regular. Every point of W is a localization of such a
closed point: take a closed point in its closure, which exists because
closures meet the closed fiber. Localizations of regular local rings
are regular, so W is regular, and the same argument gives V regular.
A regular Noetherian scheme has disjoint irreducible components, and
every component of W meets the closed fiber of W -> Spec A, which is X
and is integral. So W is integral; the same holds for V with closed
fiber D.
(v) W_0 -> Spec S is separated and quasi-compact, so flat base change
applied to the Cech complex of a finite affine cover gives
O(W) = O(W_0) (tensor over S) A = T (tensor over S) A = R.
QED
2.3 The ring is a complete normal local domain
Proposition 2.3. R is a 3-dimensional complete Noetherian normal local
domain containing C, with residue field C, and R is a finite A-module.
Proof. (a) T is a domain. Trivialize O_X(H) rationally by a nonzero
rational section tau. Dividing a section of O_X(jH) by tau^j embeds
each graded piece in the field C(X) of rational functions, and the
multiplication is compatible, so T embeds in C(X)[t, t^{-1}], which is
a domain.
(b) T is a finite graded S-module. By the projection formula the
graded piece T_j is H^0(D, F(j)), where F = f_*O_X is a coherent sheaf
on D. Choose a surjection O_D(-a)^p -> F, with kernel K. For j >= a
the sequence
0 -> K(j) -> O_D(j-a)^p -> F(j) -> 0
has H^1(D, O_D(j-a)) = 0, so H^0(D, F(j)) is the image of H^0(D,
O_D(j-a))^p = (S_{j-a})^p. That is, T_j is contained in S_{j-a} T_a.
Hence T is generated as an S-module by the finite-dimensional space
T_0 + T_1 + ... + T_a.
(c) R is the completion of T at n. Let a = (x_0,x_1,x_2)T. If the
homogeneous S-module generators of T have degrees at most a', then
a^j is contained in T_{>= j}, T_{>= j+a'} is contained in a^j,
so the a-adic and n-adic filtrations are cofinal. Therefore
R = lim T/a^j = lim T/n^j = the product of the T_j for j >= 0,
with the Cauchy product as multiplication. In particular R is
complete. Moreover T_0 = H^0(X,O_X) = C and T_{>0} = n, so the
residue field of R is C.
(d) R is a domain, finite over A. Let u = sum u_j and v = sum v_j be
nonzero, with u_j and v_k the first nonzero homogeneous parts. The
first nonzero homogeneous part of uv is u_j v_k, which is nonzero
because T is a domain. So R is a domain. Since T is a finite
S-module, R = T (tensor over S) A is a finite A-module. The map
S -> T is injective because f is surjective, and A is flat over S, so
A -> R is injective. Hence dim R = dim A = 3, and R is Noetherian
because it is finite over the Noetherian ring A.
(e) R is normal. By Lemma 2.2, W is regular and integral, so W is
normal, and O(W) = R. If a fraction of R is integral over R then it is
integral over every local ring O_{W,w}, hence lies in every O_{W,w},
hence lies in R.
QED
2.4 From a module to a sheaf on the model
Proposition 2.4. Let M be a nonzero finitely generated R-module of
depth 3. Then:
(a) M is a free A-module of some rank N > 0;
(b) M is a free R_q-module for every prime q of R different from m;
(c) the sheaf E = (p^*(M-tilde))^{**} is coherent and torsion-free on
W, agrees with M away from X, and its restriction P = E|_X is
torsion-free of positive rank r;
(d) g_*E is torsion-free and agrees with O_V^N away from D.
Proof. (a) M has depth 3 over R, so it is Cohen-Macaulay over R of
dimension 3, hence over A. A system of parameters of A is therefore an
M-regular sequence, in particular x_0,x_1,x_2 is. By the
Auslander-Buchsbaum formula, projective dimension plus depth equals the
depth of the ring, so M has projective dimension 0 over the regular
ring A: M is free.
(b) Let q be a prime of R other than m. The localization inequality
gives depth_{R_q} M_q >= depth_R M - dim(R/q), and since depth_R M = 3
and dim R = 3 this is at least 3 - dim(R/q). Concatenating a chain of
primes below q with one above q gives dim R >= dim(R/q) + dim R_q, so
3 - dim(R/q) >= dim R_q. Thus depth_{R_q} M_q >= dim R_q. By Lemma
2.2 the local ring R_q is regular, so another application of
Auslander-Buchsbaum shows that M_q is free.
(c) By (b), M-tilde is a vector bundle on the punctured spectrum of
Spec R, which p identifies with W - X. So p^*(M-tilde) is a vector
bundle there and E is reflexive, agreeing with M away from X. On a
regular integral scheme of dimension 3 a reflexive sheaf has depth at
least 2 at every point: dualizing a finite presentation writes the
sheaf as the kernel of a map between locally free sheaves, and the
depth lemma applies.
Let x be a point of X with dim O_{W,x} >= 2. The local equation t of X
is a nonzerodivisor on E because E is torsion-free, so depth P_x >= 2-1
= 1; that is, P is torsion-free there. At the generic point of X the
ring O_{W,x} is a discrete valuation ring, E_x is free of some positive
rank r, and t is again a nonzerodivisor, so P_x is free of rank r.
Hence P is torsion-free of rank r > 0.
(d) g_*E is a subsheaf of the constant sheaf of the function field of
V, so it is torsion-free. Off D, g is an isomorphism onto V - D, and E
is trivial there of rank N by (a), so g_*E agrees with O_V^N away from
D.
QED